# Properties

 Label 78.3.l.c Level $78$ Weight $3$ Character orbit 78.l Analytic conductor $2.125$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,3,Mod(7,78)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 11]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("78.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 78.l (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$2.12534606201$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876$$ x^8 - 2*x^7 + 2*x^6 + 82*x^5 + 5053*x^4 - 6736*x^3 + 6728*x^2 + 275384*x + 5635876 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{5} + 2 \beta_{2}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{4} + ( - \beta_{6} - \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} + 1) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_1 + 1) q^{7} + (2 \beta_{2} + 2) q^{8} + ( - 3 \beta_{4} - 3) q^{9}+O(q^{10})$$ q + (-b5 - b4) * q^2 + (-b5 + 2*b2) * q^3 + (-2*b5 + 2*b2) * q^4 + (-b6 - b3 + b2) * q^5 + (b5 - b4 + b2 + 1) * q^6 + (-b7 - b5 - b1 + 1) * q^7 + (2*b2 + 2) * q^8 + (-3*b4 - 3) * q^9 $$q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{5} + 2 \beta_{2}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{4} + ( - \beta_{6} - \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} + 1) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_1 + 1) q^{7} + (2 \beta_{2} + 2) q^{8} + ( - 3 \beta_{4} - 3) q^{9} + (\beta_{7} - \beta_{6} + 2 \beta_{5}) q^{10} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{2} + 2) q^{11} + ( - 4 \beta_{4} - 2) q^{12} + (\beta_{7} + \beta_{5} + 8 \beta_{4} + \beta_{3} - 6 \beta_{2} + 4) q^{13} + (\beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{14} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{15} - 4 \beta_{4} q^{16} + (\beta_{6} - 7 \beta_{5} + 7 \beta_{4} + 7 \beta_{2} - \beta_1 - 7) q^{17} + (3 \beta_{2} - 3) q^{18} + (\beta_{7} + 5 \beta_{5} + 7 \beta_{4} - \beta_{3} - 7 \beta_{2} - \beta_1 + 6) q^{19} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 2) q^{20} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 2) q^{21} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{22} + (16 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{23} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{2} - 4) q^{24} + ( - \beta_{7} + \beta_{6} + 6 \beta_{4} + 2 \beta_{3} - 26 \beta_{2} + \beta_1 + 1) q^{25} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + 8 \beta_{4} - 9 \beta_{2} + 2 \beta_1 + 7) q^{26} + (6 \beta_{5} - 3 \beta_{2}) q^{27} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} + 2) q^{28} + (\beta_{7} + \beta_{6} + 3 \beta_{5} - 11 \beta_{4} + 3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 3) q^{29} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{30} + ( - 2 \beta_{7} - 17 \beta_{5} + 17 \beta_{4} + 21 \beta_{2} - \beta_1 - 4) q^{31} + ( - 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{2} - 4) q^{32} + (4 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2}) q^{33} + (\beta_{6} + 14 \beta_{5} + 14 \beta_{4} - \beta_{3} - \beta_1 + 15) q^{34} + (2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 50 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \cdots - 48) q^{35}+ \cdots + ( - 6 \beta_{5} - 6 \beta_{4} - 6 \beta_1 - 6) q^{99}+O(q^{100})$$ q + (-b5 - b4) * q^2 + (-b5 + 2*b2) * q^3 + (-2*b5 + 2*b2) * q^4 + (-b6 - b3 + b2) * q^5 + (b5 - b4 + b2 + 1) * q^6 + (-b7 - b5 - b1 + 1) * q^7 + (2*b2 + 2) * q^8 + (-3*b4 - 3) * q^9 + (b7 - b6 + 2*b5) * q^10 + (-2*b7 + 2*b5 - 2*b2 + 2) * q^11 + (-4*b4 - 2) * q^12 + (b7 + b5 + 8*b4 + b3 - 6*b2 + 4) * q^13 + (b6 - 2*b5 + b3 + b2 - b1) * q^14 + (b7 + b5 - b4 - 2*b2 + 2*b1 - 2) * q^15 - 4*b4 * q^16 + (b6 - 7*b5 + 7*b4 + 7*b2 - b1 - 7) * q^17 + (3*b2 - 3) * q^18 + (b7 + 5*b5 + 7*b4 - b3 - 7*b2 - b1 + 6) * q^19 + (2*b7 + 2*b5 - 2*b4 - 2*b2 + 2*b1 - 2) * q^20 + (b6 - b5 - b4 - b3 + 2*b2 + 2) * q^21 + (-2*b7 - 2*b5 - 2*b4 + 2*b3 - 2*b2 - 2*b1 - 4) * q^22 + (16*b5 - 2*b4 - 2*b3 + 2*b1 - 2) * q^23 + (-2*b5 - 2*b4 + 4*b2 - 4) * q^24 + (-b7 + b6 + 6*b4 + 2*b3 - 26*b2 + b1 + 1) * q^25 + (b7 + b6 - 2*b5 + 8*b4 - 9*b2 + 2*b1 + 7) * q^26 + (6*b5 - 3*b2) * q^27 + (2*b6 - 2*b5 + 2*b2 + 2) * q^28 + (b7 + b6 + 3*b5 - 11*b4 + 3*b3 - 6*b2 + 3*b1 - 3) * q^29 + (-b7 - 2*b6 + 2*b4 - b3 + b1 - 1) * q^30 + (-2*b7 - 17*b5 + 17*b4 + 21*b2 - b1 - 4) * q^31 + (-4*b5 - 4*b4 + 4*b2 - 4) * q^32 + (4*b6 - 2*b5 + 4*b4 + 2*b3 + 4*b2) * q^33 + (b6 + 14*b5 + 14*b4 - b3 - b1 + 15) * q^34 + (2*b7 - 2*b6 + 4*b5 - 50*b4 - 2*b3 + 4*b2 - 48) * q^35 + 6*b5 * q^36 + (5*b7 + 18*b5 + 4*b4 - 22*b2 + 22) * q^37 + (2*b7 + 4*b4 - 2*b3 - 12*b2 + 4) * q^38 + (-2*b7 - 2*b6 - 13*b5 + 7*b4 - b3 + 2*b2 - b1 + 12) * q^39 + (-2*b6 - 2*b3 + 2*b1 - 2) * q^40 + (-3*b7 - 2*b6 - 7*b5 + 5*b4 + 12*b2 - 6*b1 + 12) * q^41 + (-b7 - b6 - b5 - 3*b4 - 2*b3 + 2*b2 - 2*b1 + 2) * q^42 + (b7 - b6 - 21*b5 + 18*b4 + b3 + 21*b2 + 2*b1 - 19) * q^43 + (4*b6 - 4*b5 + 4*b4 + 4*b3 + 4*b2 - 4) * q^44 + (3*b5 + 3*b4 + 3*b3 - 3*b2) * q^45 + (-2*b7 - 4*b6 + 18*b5 - 14*b4 - 2*b3 - 14*b2 - 2*b1 - 16) * q^46 + (-2*b6 - 6*b5 - 6*b4 + 2*b3 + 42*b2 + 34) * q^47 + (4*b5 + 4*b2) * q^48 + (-2*b7 + 2*b6 - 2*b5 + 4*b4 + 3*b3 - 3*b1 + 5) * q^49 + (-3*b7 + b6 - 23*b5 + 29*b4 + 2*b3 - 6*b2 + 4) * q^50 + (b7 - b6 - 14*b4 - 2*b3 - 21*b2 - b1 - 5) * q^51 + (-2*b7 - 2*b6 - 10*b5 + 12*b4 - 2*b3 - 6*b2 + 12) * q^52 + (3*b7 - b6 + 34*b5 + 2*b3 - 17*b2 + b1 + 11) * q^53 + (3*b5 - 3*b4 - 6*b2 - 6) * q^54 + (-2*b7 - 2*b6 - 8*b5 - 96*b4 - 4*b3 + 16*b2 - 4*b1 + 4) * q^55 + (-2*b7 - 2*b5 - 2*b4 - 2*b3 + 2*b2 - 2*b1 + 4) * q^56 + (2*b7 - 3*b6 - 12*b5 + 12*b4 - 3*b3 + 3*b2 + b1 + 12) * q^57 + (-3*b7 - 14*b5 - 8*b4 + b3 + 8*b2 + 3*b1 - 15) * q^58 + (-2*b7 - 8*b6 + 2*b5 + 12*b4 - 4*b3 + 12*b2 - 2*b1 + 2) * q^59 + (-2*b6 + 4*b5 + 4*b4 + 2*b3 - 2*b2) * q^60 + (3*b7 + 4*b6 + 15*b5 - 28*b4 - 3*b3 + 15*b2 + 7*b1 - 25) * q^61 + (-b7 + b6 + 25*b5 + 17*b4 + 2*b3 - 2*b1 + 32) * q^62 + (3*b7 - 3*b4 + 3*b2 - 3) * q^63 + 8*b2 * q^64 + (-b7 + 4*b6 - 55*b5 + 39*b4 - 4*b3 + 54*b2 - 6*b1 + 8) * q^65 + (-4*b7 + 2*b6 + 4*b5 - 2*b3 - 2*b2 - 2*b1 + 8) * q^66 + (b7 - 2*b6 - 32*b5 + 13*b4 + 45*b2 + 2*b1 + 45) * q^67 + (14*b5 - 14*b4 - 2*b3 - 28*b2 - 2*b1 + 2) * q^68 + (4*b7 + 2*b6 + 6*b5 + 16*b4 + 4*b3 - 6*b2 + 2*b1 - 20) * q^69 + (4*b7 - 2*b6 + 8*b5 - 8*b4 - 2*b3 + 46*b2 + 2*b1 - 52) * q^70 + (-4*b7 - 34*b5 - 12*b4 - 2*b3 + 12*b2 + 4*b1 - 32) * q^71 + (6*b5 - 6*b4 - 6*b2 - 6) * q^72 + (-b6 - 29*b5 - 29*b4 + b3 + 16*b2 + 5*b1 - 14) * q^73 + (5*b7 - 22*b5 - 14*b4 - 5*b3 - 22*b2 + 5*b1 - 9) * q^74 + (-3*b7 + 3*b6 - 9*b5 + 26*b4 + 3*b3 - 3*b1 + 49) * q^75 + (2*b7 - 2*b6 - 10*b5 + 14*b4 - 4*b3 - 4*b2 + 8) * q^76 + (-6*b7 - 4*b6 + 12*b4 + 2*b3 - 100*b2 - 4*b1 + 4) * q^77 + (b7 - 15*b5 + 7*b4 + 3*b3 + 6*b2 - b1 + 17) * q^78 + (-5*b7 + 68*b5 - 5*b3 - 34*b2 - 12) * q^79 + (-4*b6 + 4*b5 + 4*b4) * q^80 + 9*b4 * q^81 + (5*b7 + 6*b6 - 2*b5 - 12*b4 + 5*b3 + 2*b2 - b1 + 7) * q^82 + (4*b7 + 10*b6 + 40*b5 - 40*b4 + 10*b3 - 24*b2 + 2*b1 - 26) * q^83 + (2*b7 - 2*b5 - 4*b4 + 4*b2 - 2*b1 - 2) * q^84 + (9*b7 + 12*b6 + 55*b5 - 17*b4 + 6*b3 - 17*b2 + 9*b1 - 61) * q^85 + (-b6 + 36*b5 + 36*b4 + b3 + 3*b2 + 3*b1 + 38) * q^86 + (-5*b7 + b6 + 11*b5 + 9*b4 + 5*b3 + 11*b2 - 4*b1 + 4) * q^87 + (-4*b7 + 4*b6 + 4*b5 + 4*b4 + 8) * q^88 + (b7 - b6 + 10*b5 + 68*b4 - 2*b3 - 78*b2 + 80) * q^89 + (3*b6 + 3*b3 - 6*b2 + 3*b1 - 3) * q^90 + (-2*b7 + b6 + 7*b5 + 56*b4 + 6*b3 + 29*b2 + 6*b1 - 5) * q^91 + (4*b7 + 8*b5 + 4*b3 - 4*b2 - 36) * q^92 + (3*b6 - 13*b5 - 38*b4 - 25*b2 - 25) * q^93 + (2*b7 + 2*b6 - 6*b5 - 78*b4 + 4*b3 + 12*b2 + 4*b1 - 4) * q^94 + (-4*b7 + 46*b5 + 50*b4 - 4*b3 - 46*b2 - 4*b1 - 46) * q^95 + (8*b5 - 8*b4 - 4*b2 - 4) * q^96 + (4*b7 - 35*b5 + 40*b4 - b3 - 40*b2 - 4*b1 - 34) * q^97 + (-b7 + 6*b6 - 6*b5 - 2*b4 + 3*b3 - 2*b2 - b1 + 3) * q^98 + (-6*b5 - 6*b4 - 6*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} - 6 q^{5} + 12 q^{6} + 10 q^{7} + 16 q^{8} - 12 q^{9}+O(q^{10})$$ 8 * q + 4 * q^2 - 6 * q^5 + 12 * q^6 + 10 * q^7 + 16 * q^8 - 12 * q^9 $$8 q + 4 q^{2} - 6 q^{5} + 12 q^{6} + 10 q^{7} + 16 q^{8} - 12 q^{9} - 6 q^{10} + 24 q^{11} + 4 q^{14} - 12 q^{15} + 16 q^{16} - 84 q^{17} - 24 q^{18} + 10 q^{19} - 12 q^{20} + 18 q^{21} - 12 q^{22} - 12 q^{23} - 24 q^{24} + 26 q^{26} + 20 q^{28} + 36 q^{29} - 18 q^{30} - 94 q^{31} - 16 q^{32} + 60 q^{34} - 204 q^{35} + 140 q^{37} + 66 q^{39} - 24 q^{40} + 72 q^{41} + 18 q^{42} - 222 q^{43} - 24 q^{44} - 84 q^{46} + 300 q^{47} + 42 q^{49} - 62 q^{50} + 44 q^{52} + 84 q^{53} - 36 q^{54} + 396 q^{55} + 36 q^{56} + 24 q^{57} - 66 q^{58} - 60 q^{59} - 12 q^{60} - 90 q^{61} + 198 q^{62} - 24 q^{63} - 108 q^{65} + 72 q^{66} + 304 q^{67} + 60 q^{68} - 216 q^{69} - 408 q^{70} - 192 q^{71} - 24 q^{72} + 16 q^{73} - 46 q^{74} + 312 q^{75} - 20 q^{76} + 114 q^{78} - 96 q^{79} - 24 q^{80} - 36 q^{81} + 114 q^{82} - 12 q^{84} - 390 q^{85} + 168 q^{86} + 30 q^{87} + 72 q^{88} + 354 q^{89} - 218 q^{91} - 288 q^{92} - 42 q^{93} + 300 q^{94} - 576 q^{95} - 460 q^{97} + 58 q^{98} - 36 q^{99}+O(q^{100})$$ 8 * q + 4 * q^2 - 6 * q^5 + 12 * q^6 + 10 * q^7 + 16 * q^8 - 12 * q^9 - 6 * q^10 + 24 * q^11 + 4 * q^14 - 12 * q^15 + 16 * q^16 - 84 * q^17 - 24 * q^18 + 10 * q^19 - 12 * q^20 + 18 * q^21 - 12 * q^22 - 12 * q^23 - 24 * q^24 + 26 * q^26 + 20 * q^28 + 36 * q^29 - 18 * q^30 - 94 * q^31 - 16 * q^32 + 60 * q^34 - 204 * q^35 + 140 * q^37 + 66 * q^39 - 24 * q^40 + 72 * q^41 + 18 * q^42 - 222 * q^43 - 24 * q^44 - 84 * q^46 + 300 * q^47 + 42 * q^49 - 62 * q^50 + 44 * q^52 + 84 * q^53 - 36 * q^54 + 396 * q^55 + 36 * q^56 + 24 * q^57 - 66 * q^58 - 60 * q^59 - 12 * q^60 - 90 * q^61 + 198 * q^62 - 24 * q^63 - 108 * q^65 + 72 * q^66 + 304 * q^67 + 60 * q^68 - 216 * q^69 - 408 * q^70 - 192 * q^71 - 24 * q^72 + 16 * q^73 - 46 * q^74 + 312 * q^75 - 20 * q^76 + 114 * q^78 - 96 * q^79 - 24 * q^80 - 36 * q^81 + 114 * q^82 - 12 * q^84 - 390 * q^85 + 168 * q^86 + 30 * q^87 + 72 * q^88 + 354 * q^89 - 218 * q^91 - 288 * q^92 - 42 * q^93 + 300 * q^94 - 576 * q^95 - 460 * q^97 + 58 * q^98 - 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 4081829 \nu^{7} + 125878448 \nu^{6} - 175318185 \nu^{5} - 167602931 \nu^{4} - 13767350850 \nu^{3} + 914252484693 \nu^{2} + \cdots - 561922481148 ) / 28390156409030$$ (-4081829*v^7 + 125878448*v^6 - 175318185*v^5 - 167602931*v^4 - 13767350850*v^3 + 914252484693*v^2 - 589968133288*v - 561922481148) / 28390156409030 $$\beta_{3}$$ $$=$$ $$( - 900565 \nu^{7} + 114752803 \nu^{6} + 496632472 \nu^{5} + 5297934372 \nu^{4} + 3978860609 \nu^{3} + 326689911519 \nu^{2} + \cdots + 15143464458814 ) / 813197403460$$ (-900565*v^7 + 114752803*v^6 + 496632472*v^5 + 5297934372*v^4 + 3978860609*v^3 + 326689911519*v^2 + 1720952318288*v + 15143464458814) / 813197403460 $$\beta_{4}$$ $$=$$ $$( - 887 \nu^{7} - 2974 \nu^{6} - 42132 \nu^{5} - 34750 \nu^{4} - 2587559 \nu^{3} - 10377280 \nu^{2} - 118350522 \nu - 276718188 ) / 320442520$$ (-887*v^7 - 2974*v^6 - 42132*v^5 - 34750*v^4 - 2587559*v^3 - 10377280*v^2 - 118350522*v - 276718188) / 320442520 $$\beta_{5}$$ $$=$$ $$( 5758773899 \nu^{7} - 1375000768 \nu^{6} - 272272114360 \nu^{5} - 695422749614 \nu^{4} + 16988141226435 \nu^{3} + \cdots - 24\!\cdots\!12 ) / 19\!\cdots\!40$$ (5758773899*v^7 - 1375000768*v^6 - 272272114360*v^5 - 695422749614*v^4 + 16988141226435*v^3 + 12062579279902*v^2 - 775067199122402*v - 2461440553912712) / 1930530635814040 $$\beta_{6}$$ $$=$$ $$( 4272345 \nu^{7} - 119540717 \nu^{6} - 491845918 \nu^{5} - 5101492538 \nu^{4} + 21421095309 \nu^{3} - 342802118751 \nu^{2} + \cdots - 13671329236226 ) / 813197403460$$ (4272345*v^7 - 119540717*v^6 - 491845918*v^5 - 5101492538*v^4 + 21421095309*v^3 - 342802118751*v^2 - 1704850356072*v - 13671329236226) / 813197403460 $$\beta_{7}$$ $$=$$ $$( -2\nu^{7} - 17\nu^{6} + 16\nu^{5} + 798\nu^{4} - 6888\nu^{3} - 47339\nu^{2} - 13670\nu + 2105738 ) / 134980$$ (-2*v^7 - 17*v^6 + 16*v^5 + 798*v^4 - 6888*v^3 - 47339*v^2 - 13670*v + 2105738) / 134980
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} - 6\beta_{4} + 49\beta_{2} + \beta _1 - 3$$ b7 + b6 - 6*b4 + 49*b2 + b1 - 3 $$\nu^{3}$$ $$=$$ $$-4\beta_{7} + 51\beta_{6} - 46\beta_{5} + 46\beta_{4} + 51\beta_{3} + 43\beta_{2} - 2\beta _1 - 48$$ -4*b7 + 51*b6 - 46*b5 + 46*b4 + 51*b3 + 43*b2 - 2*b1 - 48 $$\nu^{4}$$ $$=$$ $$95\beta_{7} - 5\beta_{6} + 502\beta_{5} + 90\beta_{3} - 251\beta_{2} + 5\beta _1 - 2607$$ 95*b7 - 5*b6 + 502*b5 + 90*b3 - 251*b2 + 5*b1 - 2607 $$\nu^{5}$$ $$=$$ $$161\beta_{6} - 4400\beta_{5} - 4400\beta_{4} - 161\beta_{3} + 545\beta_{2} - 2702\beta _1 - 3694$$ 161*b6 - 4400*b5 - 4400*b4 - 161*b3 + 545*b2 - 2702*b1 - 3694 $$\nu^{6}$$ $$=$$ $$-6941\beta_{7} - 6235\beta_{6} + 31990\beta_{4} + 706\beta_{3} - 133847\beta_{2} - 6235\beta _1 + 15289$$ -6941*b7 - 6235*b6 + 31990*b4 + 706*b3 - 133847*b2 - 6235*b1 + 15289 $$\nu^{7}$$ $$=$$ $$19520 \beta_{7} - 147023 \beta_{6} + 323522 \beta_{5} - 323522 \beta_{4} - 147023 \beta_{3} - 265987 \beta_{2} + 9760 \beta _1 + 89488$$ 19520*b7 - 147023*b6 + 323522*b5 - 323522*b4 - 147023*b3 - 265987*b2 + 9760*b1 + 89488

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$1$$ $$\beta_{2} - \beta_{5}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −5.39181 + 5.39181i 5.02578 − 5.02578i −4.04651 − 4.04651i 5.41254 + 5.41254i −4.04651 + 4.04651i 5.41254 − 5.41254i −5.39181 − 5.39181i 5.02578 + 5.02578i
1.36603 0.366025i 0.866025 1.50000i 1.73205 1.00000i −6.39181 6.39181i 0.633975 2.36603i 9.23137 + 2.47354i 2.00000 2.00000i −1.50000 2.59808i −11.0709 6.39181i
7.2 1.36603 0.366025i 0.866025 1.50000i 1.73205 1.00000i 4.02578 + 4.02578i 0.633975 2.36603i −4.99932 1.33956i 2.00000 2.00000i −1.50000 2.59808i 6.97286 + 4.02578i
19.1 −0.366025 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i −5.04651 + 5.04651i 2.36603 + 0.633975i −1.34715 + 5.02764i 2.00000 + 2.00000i −1.50000 2.59808i 8.74082 + 5.04651i
19.2 −0.366025 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i 4.41254 4.41254i 2.36603 + 0.633975i 2.11510 7.89367i 2.00000 + 2.00000i −1.50000 2.59808i −7.64274 4.41254i
37.1 −0.366025 + 1.36603i −0.866025 1.50000i −1.73205 1.00000i −5.04651 5.04651i 2.36603 0.633975i −1.34715 5.02764i 2.00000 2.00000i −1.50000 + 2.59808i 8.74082 5.04651i
37.2 −0.366025 + 1.36603i −0.866025 1.50000i −1.73205 1.00000i 4.41254 + 4.41254i 2.36603 0.633975i 2.11510 + 7.89367i 2.00000 2.00000i −1.50000 + 2.59808i −7.64274 + 4.41254i
67.1 1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i −6.39181 + 6.39181i 0.633975 + 2.36603i 9.23137 2.47354i 2.00000 + 2.00000i −1.50000 + 2.59808i −11.0709 + 6.39181i
67.2 1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i 4.02578 4.02578i 0.633975 + 2.36603i −4.99932 + 1.33956i 2.00000 + 2.00000i −1.50000 + 2.59808i 6.97286 4.02578i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.3.l.c 8
3.b odd 2 1 234.3.bb.d 8
13.c even 3 1 1014.3.f.h 8
13.e even 6 1 1014.3.f.j 8
13.f odd 12 1 inner 78.3.l.c 8
13.f odd 12 1 1014.3.f.h 8
13.f odd 12 1 1014.3.f.j 8
39.k even 12 1 234.3.bb.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.3.l.c 8 1.a even 1 1 trivial
78.3.l.c 8 13.f odd 12 1 inner
234.3.bb.d 8 3.b odd 2 1
234.3.bb.d 8 39.k even 12 1
1014.3.f.h 8 13.c even 3 1
1014.3.f.h 8 13.f odd 12 1
1014.3.f.j 8 13.e even 6 1
1014.3.f.j 8 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} - 282T_{5}^{5} + 4065T_{5}^{4} + 11916T_{5}^{3} + 38088T_{5}^{2} - 632592T_{5} + 5253264$$ acting on $$S_{3}^{\mathrm{new}}(78, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2}$$
$3$ $$(T^{4} + 3 T^{2} + 9)^{2}$$
$5$ $$T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 5253264$$
$7$ $$T^{8} - 10 T^{7} + 29 T^{6} + \cdots + 4426816$$
$11$ $$T^{8} - 24 T^{7} + \cdots + 973440000$$
$13$ $$T^{8} - 50 T^{6} + \cdots + 815730721$$
$17$ $$T^{8} + 84 T^{7} + \cdots + 4567597056$$
$19$ $$T^{8} - 10 T^{7} + 842 T^{6} + \cdots + 1763584$$
$23$ $$T^{8} + 12 T^{7} + \cdots + 17831863296$$
$29$ $$T^{8} - 36 T^{7} + \cdots + 48599084304$$
$31$ $$T^{8} + 94 T^{7} + \cdots + 33544655104$$
$37$ $$T^{8} - 140 T^{7} + \cdots + 10744151716$$
$41$ $$T^{8} - 72 T^{7} + \cdots + 370617958656$$
$43$ $$T^{8} + 222 T^{7} + \cdots + 1819196698176$$
$47$ $$T^{8} - 300 T^{7} + \cdots + 20494380893184$$
$53$ $$(T^{4} - 42 T^{3} - 2211 T^{2} + \cdots - 109128)^{2}$$
$59$ $$T^{8} + 60 T^{7} + \cdots + 15611728564224$$
$61$ $$T^{8} + \cdots + 295174493893449$$
$67$ $$T^{8} - 304 T^{7} + \cdots + 42600893548096$$
$71$ $$T^{8} + 192 T^{7} + \cdots + 1623606027264$$
$73$ $$T^{8} - 16 T^{7} + \cdots + 261324417601$$
$79$ $$(T^{4} + 48 T^{3} - 11667 T^{2} + \cdots - 8209344)^{2}$$
$83$ $$T^{8} + 682176 T^{5} + \cdots + 13865554427904$$
$89$ $$T^{8} - 354 T^{7} + \cdots + 29\!\cdots\!64$$
$97$ $$T^{8} + 460 T^{7} + \cdots + 14\!\cdots\!96$$